Exponential growth

In connection with the current Corona crisis, the term exponential growth appears again and again. It appears to have a very disturbing effect on many people, and in fact it sounds terrifying in the context of a pandemic; reason enough for us to deal with it more extensively today.


“Exponential growth (also called unlimited or free growth) describes a mathematical model for a growth process in which the stock size is always multiplied by the same factor in the same time steps.” [1]

This definition does indeed sound very dry and abstract to a non-mathematician. For this reason, we now use a well-known Indian parable around King Sher Khan and the wise man Sissa from 1256. [2]

According to legend, King Sher Khan was a very capricious person; one day he ordered his court staff to invent a game for him to get him out of his boredom.

A wise man named Sissa invented the game of chess for him, and the king was so excited about this new game that he wanted to reward him. He asked Sissa to make a wish.

At first, he didn't want to, but when the king urged, Sissa replied that he wanted a grain of rice on the first square, two on the second, four on the third, and so on on the chessboard. At first the king was annoyed by the modesty of this wish, but the court mathematicians quickly rushed to him and calculated that he could not fulfill this wish: in fact, the number of grains of rice in the 64th field amounts to 9,223. 372,036,854,775,808 which corresponded to a multiple of the worldwide rice production.

In this case the factor is 2 because the number of rice grains per field is multiplied by the number 2:

Field Number Number of grains of rice in the field
1 1 = 20
2 2 = 21
3 4 = 22
4 8 = 23
5 16 = 24
64 9.223.372.036.854.775.808 = 263

The chessboard legend of King Shirhan

If we represent this in a graphic, we get:


So we see: an exponential growth represents a very strong growth, which also accelerates indefinitely.

This factor does not have to be exactly 2; exponential growth is generally used when this factor is greater than 1, e.g. 3 or 4 or 1.5.



[1] https://de.wikipedia.org/wiki/Exponentielles_Wachstum
[2] https://meinstein.ch/math/reis-auf-dem-schachbrett